Salam sister
There's two key verses to consider here:
The first one points to the moral and spiritual equality of men and women '...And they (the women) have rights similar and equal to those (of men) over them in equity...' (2:228)
'Men are the protectors and maintainers of women with what Allah has favoured one over the other' (4:34)
The second Ayah is referring to the general physical strength (as scientifically proven) that God has favoured men with over women.
Through that, it's ordained that a man should protect his wife physically from harm and provide for her (maintain her) economically.
That's not to say you can't work a full or part time job that you want to do and it's not to say that you can't contribute to household finances if you want to, but you should know that you have no obigation to do that if you don't want to. If you do and your husband accepts that then it's a voluntary kindness from you and your husband should be appreciative. Furthermore, your income from your job is yours alone while his income belongs to you and him and he's obliged to spend on you from it.
As regards housework and houseduties there is nothing wrong with your husband helping you with housework or even you sharing housework equally between you. However, if he is working full time and you are not working or working part time you should consider how fair this is on him. If he does go above and beyond to do that then you should be appreciative of him and acknowledge it as a kindness from him. (Also a Sunnah of the Prophet pbuh
)
If it really bothers you then an easy way to get past this issue is to hire a domestic helper to do housework for you on a weekly basis which shouldn't be too difficult to commit to if you really don't like housework and want to work instead. As for whether your husband should stay at home and do these homemaking activities without working... I would strongly not advise that unless he has some ailment or other that prevents him from working as this would go against the second verse mentioned here.
Allah knows best